Existence and Multiplicity results for Weakly coupled system of Pucci's extremal operator

In this work, we investigate the existence of multiple positive solutions for a weakly coupled system of nonlinear elliptic equations governed by Pucci extremal operators. Specifically, we consider the system: $$\begin{cases} -{M}_{\lambda_1,\Lambda_1}^+(D^2u_1) = \mu f_1(u_1, u_2, \dots, u_n), & \text{in } \Omega, \\ -{M}_{\lambda_2,\Lambda_2}^+(D^2u_2) = \mu f_2(u_1, u_2, \dots, u_n), & \text{in } \Omega, \vdots \\ -{M}_{\lambda_n,\Lambda_n}^+(D^2u_n) = \mu f_n(u_1, u_2, \dots, u_n), & \text{in } \Omega, \\ u_1 = u_2 = \dots = u_n = 0, & \text{on } \partial\Omega, \end{cases}$$ where ${M}_{\lambda,\Lambda}^+$ represents the Pucci extremal operator, $\Omega$ is a bounded domain in $\mathbb{R}^N$ with smooth boundary, and the nonlinear functions $f_i: [0, \infty)^n \to [0, \infty)$ belong to the $C^{1,\alpha}$ class. Our main results establish the existence and multiplicity of solutions for sufficiently large values of the parameter $\mu > 0$. The analysis relies on the method of sub and supersolutions, in conjunction with fixed-point arguments and bifurcation techniques.